Quadratic Equations and Inequalities


Quadratic Equations and Inequalities


\(\blacksquare\) A quadratic equation in general form can be written as


where \(a\), \(b\) and \(c\) are constants and \(a \ne 0\)


\(\blacksquare\) Methods used to solve quadratic equations:

1. Completing the square

2. Formula

\(\boxed{x=\dfrac{-b \pm \sqrt{b^2-4ac}} {2a}}\)


\(\blacksquare\) If \(\alpha\) and \(\beta\) are the roots of a quadratic equation, then





\(\blacksquare\) For \(x^2-(\alpha+\beta)x+\alpha\beta=0\),

\(\alpha+\beta\) is the sum of roots and \(\alpha\beta\) is the product of roots.


\(\blacksquare\) For a quadratic equation in the form


where \(a \lt b\),

  • if \((x-a)(x-b)\gt0\), then \(x \lt a\) or \(x \gt b\).
  • if \((x-a)(x-b)\lt0\), then \(a\lt x \lt b\).